Unimodular bilinear Fourier multipliers on $L^p$ spaces

In this paper we investigate the boundedness properties of bilinear multiplier operators associated with unimodular functions of the form $m(ξ,η)=e^{i ϕ(ξ-η)}$. We prove that if $ϕ$ is a $C^1(\mathbb R^n)$ real-valued non-linear function, then for all exponents $p,q,r$ lying outside the local $L^2-$range and satisfying the Hölder's condition $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, the bilinear multiplier norm $\|e^{iλϕ(ξ-η)}\|_{\mathcal M_{p,q,r}(\mathbb R^n)}\rightarrow \infty,~ λ\in \mathbb R,~ |λ|\rightarrow \infty.$ For exponents in the local $L^2-$range, we give examples of unimodular functions of the form $e^{iϕ(ξ-η)}$, which do not give rise to bilinear multipliers. Further, we also discuss the essential continuity property of bilinear multipliers for exponents outside local $L^2-$ range.

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